Existence Of Solutions For A Class Of Fractional Elliptic Problems With Critical Growth In Unbounded Domains

This dissertation mainly studies the existence of solutions for the following class of non-local elliptic problems(?)where RN\? is bounded,?>0,s?(0,1),N>2s,p ?(2,2s*),2s*=2N/N-2s is the fractional critical Sobolev exponent.First of all,in the case of ?=RN.with the help of Nehari manifold,we prove that this problem has a ground state solution when ??(0,?0)for some?0 small enough.Then we also consider the situation when ? is an exterior domain with non-empty smooth boundary and prove that the above equation does not have a ground state solution when ??(0,?0).Furthermore,when ? is an exterior domain and satisfies RN\?(?)B(0,p)with p is a sufficiently small positive number,we use the results in Alves et al.[J.Differ.Equations,2020]to overcome the difficulty of the lack of compactness due to the unboundness of ? and prove that the equation has at least one non-trivial solution when ? is sufficiently small by combining with the theory of topological degree and deformation lemma,which extends the main results of Alves et al.[Milan.J.Math.,2017]to non-local cases.